2 - Basics of fractional calculus and fractional order differential equations

Abstract

This chapter introduces first a brief survey of the history of fractional calculus and fractional order differential (more exactly integro-differential) equations. We consider Riemann–Liouville fractional integrals and derivatives on a segment and on a semiaxis, Gerasimov–Caputo, Dzrbashian–Nersesyan, sequential, averaged, and distributed order fractional derivatives, Erdélyi–Kober, Saigo, Love, and other fractional operators, and fractional integrals and fractional derivatives of a function with respect to another function. After that integral transforms and basic differential equations of fractional order are considered. We also emphasize the impact on applications of fractional calculus of Soviet and Russian mechanics, including those from the Voronezh school.